Quantum Optics of Soliton Microcombs ,1 ,1 ,1 1,2 ,1 Melissa A. Guidry∗ , Daniil M. Lukin∗ , Ki Youl Yang∗ , Rahul Trivedi and Jelena Vuˇckovi´c† 1E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA. 2Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany Soliton microcombs — phase-locked microcavity frequency combs — have become the foundation of several classical technologies in integrated photonics, including spectroscopy, LiDAR, and optical computing. Despite the predicted multimode entanglement across the comb, experimental study of the quantum optics of the soliton microcomb has been elusive. In this work, we use second- order photon correlations to study the underlying quantum processes of soliton microcombs in an integrated silicon carbide microresonator. We show that a stable temporal lattice of solitons can isolate a multimode below-threshold Gaussian state from any admixture of coherent light, and predict that all-to-all entanglement can be realized for the state. Our work opens a pathway toward a soliton-based multimode quantum resource. Kerr optical frequency combs1–3 are multimode states of light generated via a third-order optical nonlinearity in ˆ g0 an optical resonator. When a Kerr resonator is pumped Hint = δFWM(AmAnaˆ†aˆ† +A∗Anaˆ†aˆm +h.c.) − 2 j k k j weakly, spontaneous parametric processes populate res- m,n,j,kX onator modes in pairs. In this regime, Kerr combs can be (1) a quantum resource for the generation of heralded single where δFWM = δ(j+k m n) is the four-wave mixing 4–6 − − photons and energy-time entangled pairs , multiphoton (FWM) mode number matching condition and g0 is entangled states7–9, and squeezed vacuum10–12. When the nonlinear coupling coefficient. This approximation pumped more strongly, the parametric gain can exceed has successfully modelled the quantum correlations de- the resonator loss and give rise to optical parametric os- veloped by a coherent soliton pulse after propagating cillation (OPO) and bright comb formation. The modes through a fiber27,28, where the Kerr interaction is weak. of a Kerr comb can become phase-locked to form a stable, The DKS state, however, is itself generated by strong low-noise dissipative Kerr soliton (DKS). This regime of Kerr interactions where linearization can break down and Kerr comb operation has become the foundation of mul- yield unphysical predictions29,30. Experimentally verify- tiple technologies, including comb-based spectroscopy13, ing the validity of this model for the DKS is a prerequisite LiDAR14, optical frequency synthesizers15, and optical for exploring its application in quantum technologies. processors16. In this work, we use second-order photon correlation measurements to experimentally validate the linearized While the soliton microcomb has been mostly stud- model (Eq. 1) for describing the spontaneous parametric ied classically, it is nonetheless fundamentally governed processes in DKS states. We study the quantum for- by the dynamics of quantized parametric processes: each mation dynamics of Kerr frequency combs and identify resonator mode is coupled to every other mode through a class of DKS states — perfect soliton crystals31,32 — a four-photon interaction. The multimode coupled pro- which naturally isolate a subset of quantum optical fields cesses of the DKS resemble those of the well-studied 17,18 from the coherent mean-field (Fig. 1). We measure the synchronously pumped OPO . If the quantum pro- correlation matrix of the below-threshold modes to nu- cesses can be harnessed, soliton microcombs may open merically infer the multi-mode entanglement structure of a pathway toward the experimental realization of a 18–20 the state. We find that the DKS state may be engineered multimode quantum resource in a scalable, chip- as an on-chip source of all-to-all entanglement. integrated platform21,22. However, the quantum-optical properties of soliton microcombs have not yet been di- The observation of the full optical spectrum of a rectly observed: the first glimpse into non-classicality was Kerr frequency comb, including the above- and below- the recent measurement of the quantum-limited timing threshold processes, requires single-photon sensitivity arXiv:2103.10517v1 [physics.optics] 18 Mar 2021 jitter of the DKS state23,24. and a high dynamic range. For this purpose, we designed a single-photon optical spectrum analyzer (SPOSA) Linearization25 — the formal separation of the opti- using multipass grating monochromators and super- cal state into the mean field solution and the quantum conducting nanowire single-photon detectors (SNSPDs) fluctuations — is one approach to simplify the otherwise (PhotonSpot, Inc). The SPOSA has a broadband intractable many-body quartic Hamiltonian of the DKS. (> 200 nm) quantum efficiency of 20%, and close-in The result of this approximation is a quadratic Hamilto- dynamic range of > 140 dB at 0∼.8 nm. We generate ± nian, where the classical amplitudes Am, obtained for ex- Kerr frequency combs using microring resonators with ample through the spatiotemporal Lugiato-Lefever equa- 350 GHz free spectral range (FSR) fabricated in 4H- tion (LLE)26, drive the spontaneous parametric processes silicon carbide-on-insulator, a platform with favorable 33 34 between quantum modesa ˆj: nonlinear and quantum optical properties. With in- 2 FIG. 1: Linearized model for quantum optical fields in a DKS state A schematic depiction of a Kerr mi- croresonator with a circulating perfect soliton crystal state. The full optical state is modeled as a coherent classical comb (blue) that drives the quantum comb (red) via spontaneous parametric processes. FIG. 2: Single-photon spectroscopy of optical microcombs (a) Stages of frequency comb formation observed on the single-photon optical spectrum analyzer (SPOSA). Dashed line indicates noise floor of a commercial opti- cal spectrum analyzer (-80 dBm). (b) The single soliton state. (c) A 7-FSR soliton crystal state, observed in a different device. trinsic quality factors (Q) as high as 5.6 106, low thresh- 3 modes with very low photon number aj†aj < 10− . old OPO and low-power soliton operation× (0.5 mW and The quantum fluctuations generated inD theE DKS state 2.3 mW, respectively) are achieved. We use the SPOSA and responsible for its quantum-limited timing jitter23,24 to characterize the Kerr microcomb spectra through all 2 are obscured in the single-soliton spectrum. A perfect stages of DKS formation (Fig. 2a,b). The sech soliton soliton crystal31,32 (henceforth referred to as a soliton envelope is seen to persist at the tails of the DKS, in crystal), however, reveals these quantum fluctuations 3 FIG. 3: Quantum coherence of parametric oscillation (a) The near-threshold spectrum reproduced from (2) Fig. 2a. Vertical dashed lines indicate the modes where the primary comb will form. (b) Measured gauto(τ) on dif- ferent modes shows the dispersion-dependent parametric gain variation throughout the comb. (c) Observation of asymptotic growth of coherence near the OPO threshold. The effective parametric gain λ, extracted from a numer- ical fit to Eq. 2 is plotted against the detected count rate on mode µ = +5. (d) At the highest photon count rate, (2) gauto(τ) reveals coherence-broadening which corresponds to threshold proximity of 0.99896(5). (Fig. 2c). cess (analogous to lasing). In a Kerr frequency comb close The formation of DKS from a below-threshold quan- to but below the OPO threshold (Fig. 3a), both regimes (2) tum frequency comb begins with the transition of a spon- can be observed simultaneously. We measure gauto(τ) taneous FWM process into a stimulated FWM process using two SNSPD detectors in a Hanbury Brown and at the onset of OPO. This transition can be observed Twiss configuration, and observe the dispersion-induced through the second-order correlation function, g(2)(τ). variation in λ for different signal-idler pairs (Fig. 3b).36. Using input-output theory35 we derive the exact form of Mode µ = +9, far away from the pump, displays oscil- (2) (2) g (τ) for a general two-mode parametric process. The lations in gauto(τ), signifying poor phase-matching. In general derivation is presented in the Supplementary In- contrast, mode µ = +5 shows a substantial coherence in- formation. For signal and idler modes of equal linewidths crease, which correctly predicts that it will seed the for- κ, the auto-correlation is given by mation of the primary comb. To observe the asymptotic coherence growth at threshold, we repeatedly sweep the κτ 2 (2) e− κ pump laser detuning through the OPO threshold con- gauto(τ) = 1 + sinh(λτ) + λ cosh(λτ) (2) λ2 2 dition, while synchronously acquiring the photon count h i rates and two-photon correlations. Through the numer- where λ = g2 δ2 is the effective parametric gain, 2 − ical fit to Eq. 2, λ is extracted and plotted against the g = g0 A0 is the mode coupling strength, and δ is the | | p 2 2 measured count rate (Fig. 3c). The maximum recorded detuning. Here, two regimes are notable: when δ > g , 35 (2) coherence broadening exceeds the cavity coherence by λ is imaginary which gives rise to oscillations in gauto(τ), nearly three orders of magnitude (Fig. 3d), which indi- corresponding to the double-peaked photon spectrum for cates that the state is approaching the critical point at 25,30 a strongly detuned parametric process ; when λ ap- which the linearized model would break down.29,30 (2) proaches κ/2, the gauto(τ) coherence (decay time) in- creases asymptotically, reflecting the transition of the In the linearized model of an above-threshold Kerr spontaneous FWM process into a stimulated FWM pro- comb, a resonator mode may be occupied by both a co- 4 FIG. 4: Formation dynamics of secondary combs (a) A SPOSA spectrum of a secondary comb state. (b) A graphical representation of the stages of secondary comb formation.